Use P Q R with right angle R to find sinP, cosP, tanP sinQ,...

Use $\triangle P Q R$ with right angle $R$ to find $\sin P, \cos P, \tan P$
$\sin Q, \cos Q,$ and tan $Q .$ Express each ratio as a fraction, and as a decimal to the nearest hundredth.
$$p=2 \sqrt{3}, q=\sqrt{15}, \text { and } r=3 \sqrt{3}$$

Key Concepts

Right Triangle Trigonometry

This concept involves understanding the properties of triangles that have one 90° angle. It provides the framework for defining and calculating the trigonometric ratios—sine, cosine, and tangent—based on the lengths of the sides of a right triangle.

Trigonometric Ratios

Sine, cosine, and tangent are defined as specific ratios of the side lengths in a right triangle. The sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.

Pythagorean Theorem

The Pythagorean theorem is a fundamental relationship in Euclidean geometry that relates the lengths of the sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, which is often used to find an unknown side length when two are known.

Complementary Angle Relationships

In a right triangle, the two non-right angles are complementary, meaning their measures add up to 90°. This fact establishes relationships between their trigonometric ratios, such as the sine of one angle being equal to the cosine of its complement.

Fraction and Decimal Conversion

This involves expressing numerical values in different formats for clarity and precision. In trigonometry, it is common to represent the results of ratio calculations as both exact fractions and decimal approximations to a specified degree of accuracy.

Transcript

00:01 So in this problem, we're given a right triangle and the side lengths of all three sides, and we're asked about different trigonometric properties of the angles.

00:09 And so let's go ahead and fill in the sides here.

00:12 We know that little r is 3, root 3, little p is just 2 root 3, and little q is square root of 15.

00:25 And the first thing they want to know is sine of p.

00:30 And now we can get this pretty easily since sine is just opposite over hypotenuse.

00:35 And so since p is down here, the opposite is 2 root 3.

00:40 And then the hypotenuse is just 3 or 3.

00:43 The hypotenuse is the long side of the triangle.

00:47 Square root of 3 is cancel.

00:49 And so it's just equal to 2 thirds, which is 0 .67 in decimal form.

00:56 And for all of these, i'm going to go to the second decimal place, since in the problem they asked for the nearest hundreds, which is the second decimal place.

01:05 So now let's figure out cosine.

01:07 Cosine is adjacent over hypotenuse, and the adjacent is square root of 15, like that, over hypotenuse.

01:18 And so if you simplify the square roots, you're going to get square root of 5 over 3, and then at this point we need the calculator.

01:25 So let me plug this in real fast, and i get 0 .75.

01:36 Now i need tangent of p.

01:39 Tangent is opposite over adjacent.

01:42 So this is going to be 2 or 3 over square root of 15.

01:47 Which if we simplify the square roots again, we get 2 over root 5...